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Question
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
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Solution
`int_-a^a 9^x/(1 + 9^x)dx`
Let 1 + 9x = u
On differentiating with respect to x.
`\implies` 9x . log 9 = `(du)/dx`
`\implies` 9x dx = `(du)/log9`
= `int_(1 + 9^-a)^(1 + 9^a) (du)/(log9.u)`
= `1/log9 . [log (u)]_(1 + 9^-a)^(1 + 9^a) + C`
= `1/log9 [log (1 + 9^a) - log (1 + 9^-a)] + C`
= `1/log9 [log ((1 + 9^a)/(1 + 9^-a))] + C`.
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